Global Existence and Asymptotic Behavior of Solutions of Nonlinear ...

Funkcialaj Ekvacioj, 47 (2004) 167–186

Global Existence and Asymptotic Behavior of Solutions of Nonlinear Di¤erential Equations By Octavian G. Mustafa and Yuri V. Rogovchenko* (University of Craiova, Romania and Eastern Mediterranean University, Turkey)

Abstract. The paper addresses two open problems related to global existence of solutions with a ‘‘linear-like’’ behavior at infinity. For a class of second-order nonlinear di¤erential equations, we establish global existence of solutions under milder assumption on the rate of decay of the coe‰cient. Furthermore, as opposed to results reported in the literature, we prove for another class of second-order nonlinear di¤erential equations that the region of the initial data for the solutions with desired asymptotic behavior is unbounded and proper. Key words and phrases. Nonlinear di¤erential equation, Second order, Global existence, Nonoscillatory solutions, Asymptotic behavior, Linear-like behavior at infinity, Radial solutions, Nonlinear elliptic equations. 2000 Mathematics Subject Classification Numbers. Primary 34D05; Secondary 34A12, 34A34.

1.

Introduction The second-order nonlinear ordinary di¤erential equation

ð1Þ

u 00 þ f ðt; u; u 0 Þ ¼ 0;

t b t0 ;

attracts constant interest of researchers because of its importance for mathematical modeling of di¤erent physical, chemical or biological systems. Numerous papers published recently are concerned with local and global existence of solutions of Eq. (1) and its particular cases, uniqueness, continuation, asymptotic behavior of solutions (including boundedness, prescribed asymptotic behavior, oscillation and nonoscillation), stability properties, etc. Since Eq. (1) can be viewed as a nonlinear perturbation of a very simple di¤erential equation u 00 ¼ 0 that has solutions of the form uðtÞ ¼ at þ b, numerous authors were interested in establishing su‰cient conditions for the so-called ‘‘linear-like’’ behavior of * Supported in part by the Research Fellowship in Mathematics of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

168

Octavian G. Mustafa and Yuri V. Rogovchenko

solutions at infinity referred to as ‘‘Property (L)’’ in [12]. This property reads as follows: for every continuable solution of (1) (or its particular cases) there exists a real constant a such that ð2Þ

lim u 0 ðtÞ ¼ lim

t!þy

t!þy

uðtÞ ¼ a: t

The standard assumption on the nonlinear function f ðt; u; u 0 Þ is juj ð3Þ j f ðt; u; u 0 Þj a h1 ðtÞg1 ðju 0 jÞ þ h2 ðtÞg2 þ h3 ðtÞ; t where the functions hi ðtÞ, i ¼ 1; 2; 3, are continuous, nonnegative and integrable on ½t0 ; þyÞ, while g1 ðsÞ and g2 ðsÞ are continuous, nonnegative, nondecreasing functions which satisfy certain integral conditions. Among numerous papers concerned with existence of solutions of various classes of linear and nonlinear di¤erential equations possessing property (L), we would like to refer to recent papers by Cohen [1], Constantin [2], Kusano and Trench [7, 8], Meng [9], Mustafa and Rogovchenko [10], S. Rogovchenko and Yu. Rogovchenko [12], Rogovchenko [13], Rogovchenko and Villari [14], Tong [15] and Trench [16]. The reader may consult the papers by the present authors [10] and by S. Rogovchenko and Yu. Rogovchenko [12] where further details and additional references can be retrieved. We also note that the study of linear-like solutions of second-order ordinary di¤erential equations has attracted special interest of researchers dealing with radial solutions of certain classes of elliptic equations. We address the reader to recent papers by Fukagai [4], Kusano, Naito and Usami [6], Kusano and Trench [7, 8], Usami [17], Zhang [19], and the references therein. As it has been pointed out by Kusano and Trench [8, p. 381], most results on the existence of solutions with prescribed asymptotic behavior for nonlinear equations are ‘‘local’’ near infinity, in the sense that solutions with the required behavior are shown to exist only for t large enough. Only a few papers (see, for instance, [7], [8], [10], [11]) provide global conditions which imply existence of solutions on ½t0 ; yÞ for some real t0 . Global existence of solutions of Eq. (1) with the linear-like behavior at infinity has been studied very recently by the authors [10] under the following assumption on the nonlinearity: juj j f ðt; u; vÞj a hðtÞ p1 ð4Þ þ p2 ðjvjÞ ; t where h; p1 ; p2 : ð0; þyÞ ! ð0; þyÞ are continuous, p1 and p2 are nondecreasing, h satisfies

Global Existence and Asymptotic Behavior

ð5Þ

ðy

169

hðsÞds < þy;

t0

and ð6Þ

GðþyÞ ¼ þy;

where GðxÞ ¼

ðx t0

ds : p1 ðsÞ þ p2 ðsÞ

It has been proved, among other results, that (A) for every pair of real numbers u 0 ; u1 , there exists at least one solution uðtÞ of Eq. (1) satisfying initial conditions ð7Þ

uðt0 Þ ¼ u 0 ;

u 0 ðt0 Þ ¼ u1 ;

such that uðtÞ ¼ at þ oðtÞ as t ! þy [10, Theorem 3]; (B) for every real a, there exists a solution uðtÞ of Eq. (1) defined on ½t0 ; þyÞ with the asymptotic representation uðtÞ ¼ at þ oðtÞ as t ! þy [10, Theorem 4]. This paper continues studies on global existence of solutions with linear-like behavior at infinity and is concerned with two open problems discussed in Sections 2 and 3. The first problem is related with the integrability condition (5) and its analogues in [1], [2], [9], [10], [12], [13], [15] and [16]. Is it necessary for this type of asymptotic behavior, or it is only technical? As a motivating example, consider a simple linear di¤erential equation 1 0 u 00 u þ u ð8Þ ¼ 0; t b 1: t t Clearly, assumption (5) fails to hold, although a straightforward computation shows that Eq. (8) has a two-parametric family of solutions uðtÞ ¼ C1 t þ C2 t1 defined on ½1; yÞ and possessing the property (L). The second problem is concerned with geometric properties of the region of linear-like behavior (we adopt this concept, similar to that used in stability theory, to denote the regions in the ðu; u 0 Þ-plane where initial data ðu 0 ; u1 Þ for solutions that possess property (L) are located). Should these regions be bounded as in [3], [12], and [18], or there are other possibilities? The class of equations with nonlinearities that satisfy condition (4) is very large. As it has been shown by the authors in [10, Section 5.2], condition (5) is necessary only when the information one has about Eq. (1) is the inequality

170


(4). However, if we slightly modify conditions imposed on the functions in the inequality (4), restriction (5) becomes no longer necessary and can be replaced with a milder one. Namely, in this paper we can have pi ð0Þ ¼ 0, possibility that was excluded by requiring that pi ðwÞ > 0 for all w b 0 in our paper [10], and this modification of the basic assumption on the nonlinearity f enables us to prove global existence of solutions with linear-like behavior at infinity without imposing condition (5). Furthermore, in this paper we do not require condition (6). Therefore, in general, one can establish existence of solutions of Eq. (1) that possess property (L) only for a part of solutions with initial data satisfying an additional condition. As opposed to the papers by Dannan [3], S. Rogovchenko and Yu. Rogovchenko [12], and Waltman [18], where the regions of linear-like behavior have di¤erent nature but are always bounded, the novelty of Theorem 6 also lies in the fact that for the considered class of equations the region of linear-like behavior is unbounded and proper (that is, it is neither void nor R 2 ). The model equation motivating our study is u n 00 0 u þ qðtÞ u ð9Þ ¼ 0; t b t0 ; t where n b 1 is an integer, and the function qðtÞ is continuous. elementary inequality

Using the

ðx þ yÞ 2n a ð2 maxfjxj; jyjgÞ 2n a 2 2n ðx 2n þ y 2n Þ that holds for all real numbers x and y, it is not di‰cult to check that the nonlinearity in Eq. (9) satisfies condition (4), where hðtÞ ¼ 2 2n jqðtÞj, pi ðwÞ ¼ w n , i ¼ 1; 2. We point out that not all solutions of Eq. (9) exist in the future (that is, are not defined for all t b t0 ). For instance, di¤erential equation 1 u 2 u 00 þ 2 u 0 ð10Þ ¼ 0; t b 1; t t admits a solution ð11Þ

uðtÞ ¼ 2t ln

2t ; 2þt

t A ½1; 2Þ

which blows up in finite time, as well as a one-parametric family of solutions uðtÞ ¼ Ct;

C A R;

that possess property (L) and exist on the entire interval ½1; þyÞ. This example prompts that the choice of initial data a¤ects significantly global existence and asymptotic properties of solutions of Eq. (9).


2.

171

Global solutions with linear-like behavior

In this section, we are concerned with the nonlinear di¤erential equation u 00 0 u þ aðtÞg u ð12Þ ¼ 0; t b t0 ; t def

where gðxÞ ¼ f ðxÞx 2n , the functions a : ½t0 ; þyÞ ! ½0; þyÞ and f : R ! R are continuous, xf ðxÞ > 0 for all x 0 0, t0 b 1, and n b 1 is an integer. First we note that g u 0 u a sup jgðvÞj t jvjaju 0 jþjuj=t a sup jgðvÞj þ jvja2ju 0 j

¼ p1 ðju 0 jÞ þ p2

sup jgðvÞj jvja2juj=t

juj ; t

where the functions pi ðwÞ are continuous, nondecreasing, nonnegative and pi ðwÞ > 0 for all w > 0. Therefore, the nonlinearity in Eq. (12) satisfies condition (4). Theorem 1. All solutions of Eq. (12) (a) exist in the future; (b) possess the property (2). Proof. Consider a pair of numbers u 0 ; u1 A R. Clearly, there exists a solution uðtÞ (not necessarily unique) of the di¤erential equation u u 2n 00 0 0 u þ aðtÞ f u ¼0 u t t satisfying the initial condition (7) and defined on the maximal interval ½t0 ; TÞ, where T ¼ Tu a þy. Using the transformation u 00 ðtÞ ¼ t1 ½tu 0 ðtÞ uðtÞ 0 ;

t A ½t0 ; TÞ;

we write Eq. (12) as ð13Þ

0 ðtu 0 uÞ 0 tu u tu 0 u 2n þ aðtÞ f ¼ 0; t t t

t A ½t0 ; TÞ:

Introducing the functions vðtÞ ¼ tu 0 ðtÞ uðtÞ and zðtÞ ¼ t1 vðtÞ, we write Eq. (13) first as 2n v v v 0 þ taðtÞ f ¼ 0; t A ½t0 ; TÞ; t t

172


and then in the form tz 0 þ z þ taðtÞ f ðzÞz 2n ¼ 0;

t A ½t0 ; TÞ:

We divide now the proof into the following steps. (i) First, we prove that all solutions zðtÞ of the Cauchy problem ð14Þ ð15Þ

z 0 ¼ t1 z aðtÞ f ðzÞz 2n ;

t b t0 ;

zðt0 Þ ¼ z0 ;

exist in the future. (ii) Second, we show that for any solution zðtÞ of (14), (15) one has zðtÞ ¼ Oðt1 Þ as t ! þy. To prove claim (i), let z0 ðtÞ be a solution of the Cauchy problem (14), (15) defined for t A ½t0 ; TÞ. We distinguish the following two cases. Case 1: Assume that z0 ¼ 0. It follows from (14), (15) that zðtÞ ¼ 0 is a solution of the given Cauchy problem for t b t0 . We shall prove that it is unique. Suppose, for the sake of contradiction, that there is another solution zðtÞ which is not identically zero for t b t0 . Without loss of generality, we can assume that there exists a t1 A ðt0 ; TÞ such that zðt1 Þ > 0. Then, by the continuity argument, there exists a t2 A ½t0 ; t1 Þ such that zðt2 Þ ¼ 0

and

zðtÞ > 0

for t A ðt2 ; t1 :

Since zðt2 Þ ¼ 0 < zðt1 Þ, there exists a t3 A ðt2 ; t1 Þ such that ð16Þ

zðt2 Þ ¼ 0 < zðt3 Þ < zðt1 Þ:

Using the sign property of the function f ðxÞ, we deduce that z 0 ðtÞ < 0 for t A ðt2 ; t1 . The latter inequality implies zðt3 Þ > zðt1 Þ, which contradicts (16). Thus, zðtÞ a 0 for t A ½t0 ; TÞ. Repeating the argument, we conclude that there exists no t4 A ðt0 ; TÞ such that zðt4 Þ < 0. Hence, zðtÞ ¼ z0 ðtÞ ¼ 0, t A ½t0 ; TÞ, is the only solution of the Cauchy problem (14), (15), and T ¼ þy. Case 2: Let z0 0 0. As above, all solutions of the Cauchy problem (14), (15) are defined for t A ½t0 ; TÞ, where T a þy, and can be classified in accordance with their asymptotic behavior as follows: Type (I): solutions vanishing eventually; Type (II): positive decreasing solutions; Type (III): negative increasing solutions. Clearly, all solutions of the Cauchy problem (14), (15) are bounded on their maximal intervals of existence and therefore they exist in the future. The validity of claim (i) is established. To prove claim (ii), consider z0 > 0 and assume first that z0 ðtÞ is a solution of the Cauchy problem (14), (15) of type (II), that is, z0 ðtÞ > 0 for t A ½t0 ; þyÞ.


173

Then z0 ðtÞ is the unique solution of the Cauchy problem ð17Þ ð18Þ

z 0 ¼ t1 z aðtÞ f ðz0 ðtÞÞz 2n ;

t A ½t0 ; þyÞ;

zðt0 Þ ¼ z0 > 0;

where z0 ¼ u1 t1 0 u0:

ð19Þ

Integration of Bernoulli equation (17) requires a new variable wðtÞ ¼ ½zðtÞ 12n . def

Then wðt0 Þ ¼ z012n ¼ w0 > 0 and w 0 ðtÞ ¼ ð1 2nÞz 0 ðtÞ½zðtÞ2n ¼ ð1 2nÞ½t1 zðtÞ aðtÞ f ðz0 ðtÞÞ½zðtÞ 2n ½zðtÞ2n ¼ ð2n 1Þ½t1 wðtÞ þ aðtÞ f ðz0 ðtÞÞ;

t A ½t0 ; þyÞ:

Solving the Cauchy problem w 0 ¼ ð2n 1Þ½t1 wðtÞ þ aðtÞ f ðz0 ðtÞÞ;

t A ½t0 ; þyÞ;

wðt0 Þ ¼ w0 > 0; by variation of constants method, we obtain ðt aðsÞ f ðz0 ðsÞÞ 2n1 w0 wðtÞ ¼ t þ ð2n 1Þ ds ; s 2n1 t02n1 t0

t A ½t0 ; þyÞ:

Coming back to Bernoulli equation (17), we have zðtÞ ¼ ½wðtÞ1=ð2n1Þ " 1=ð2n1Þ #1 ðt w0 aðsÞ f ðz0 ðsÞÞ ¼ t 2n1 þ ð2n 1Þ ds ; s 2n1 t0 t0

t A ½t0 ; þyÞ:

Hence, z0 ðtÞ satisfies for t A ½t0 ; þyÞ the integral equation " 1=ð2n1Þ #1 ðt w0 aðsÞ f ðz0 ðsÞÞ z0 ðtÞ ¼ t 2n1 þ ð2n 1Þ ð20Þ ds : s 2n1 t0 t0 Since z0 ðtÞ > 0 for all t b t0 , we deduce that f ðz0 ðsÞÞ > 0 for all s b t0 and 0 < z0 ðtÞ a

t0 1=ð2n1Þ w0

1 z0 t 0 ¼ t t

for t b t0 :

174


Therefore, z0 ðtÞ ¼ Oðt1 Þ

ð21Þ

as t ! þy:

Claim (ii) for solutions of type (II) is established. Consider now solutions of type (III) taking z0 < 0 and assuming that z0 ðtÞ < 0, t A ½t0 ; þyÞ, is a solution of the Cauchy problem (14), (15). Following the same lines, we conclude that z0 ðtÞ satisfies Eq. (20). Furthermore, since f ðz0 ðsÞÞ < 0 for all s b t0 , we deduce that 0 > z0 ðtÞ b

z0 t 0 t

for t b t0 ;

and thus solution z0 ðtÞ satisfies (21). Hence, claim (ii) is established for all solutions of types (II) and (III), and is evident for solutions of type (I). Let z0 ðtÞ be a solution of (14), (15), where z0 is defined by (19). Consider another Cauchy problem with the nonhomogeneous term z0 ðtÞ: u0

ð22Þ ð23Þ

u ¼ z0 ðtÞ; t

t A ½t0 ; TÞ;

uðt0 Þ ¼ u 0 :

Clearly, the unique solution uðtÞ of (22), (23) is a solution of Eq. (12) on ½t0 ; TÞ. Applying variation of constants method, we obtain for t A ½t0 ; TÞ ðt u0 z0 ðsÞ ð24Þ ds ; þ uðtÞ ¼ t s t0 t0 ðt u0 z0 ðsÞ ds þ z0 ðtÞ: u 0 ðtÞ ¼ þ ð25Þ s t0 t0 Eqs. (24) and (25) help to establish easily the following two facts. (a) Assuming, for the sake of contradiction, that T < þy and using the estimate sup ½ju 0 ðtÞj þ juðtÞj a sup jz0 ðtÞj t A ½t0 ; TÞ

t A ½t0 ; TÞ

ju 0 j þ þ ð1 þ TÞ t0

ðT t0

jz0 ðsÞj ds < þy; s

we conclude that T ¼ þy. (b) Using the fact that z0 ðtÞ ¼ Oðt1 Þ as t ! þy, we deduce that the improper integral ðy z0 ðsÞ ds s t0


175

converges, and thus u0 lim u ðtÞ ¼ þ t!þy t0 0

ðy t0

z0 ðsÞ ds A R: s

Therefore, all solutions of Eq. (12) exist in the future and possess property (2), which completes the proof. 9 Remark 2. It follows from (20) that the only solution of Eq. (14) of type (I) is the trivial one. In fact, suppose to the contrary that z0 ðtÞ is a solution of type (I) such that for some t1 one has z0 ðtÞ > 0, t A ½t0 ; t1 Þ and z0 ðtÞ ¼ 0 for all t b t1 . Then z0 ðtÞ satisfies on ½t0 ; t1 Þ Eq. (20) and z0 ðt1 Þ ¼ 0 ¼ lim z0 ðtÞ t!t1

" 1=ð2n1Þ #1 ð t1 w0 aðsÞ f ðz0 ðsÞÞ ¼ t1 2n1 þ ð2n 1Þ ds > 0; s 2n1 t0 t0 a contradiction. Example 3. ð26Þ

Consider the nonlinear di¤erential equation

u 00 þ aðt2 þ C 2 t 2a Þ

ðtu 0 uÞ 3 t 2 þ ðtu 0 uÞ 2

¼ 0;

t b 1;

where C 0 0 and a > 0 are real constants. Here n ¼ 1, f ðzÞ ¼ zð1 þ z 2 Þ1 , aðtÞ ¼ aðt1 þ C 2 t 2aþ1 Þ, and t0 ¼ 1. It is not di‰cult to verify all conditions of Theorem 1. In fact, Eq. (26) has the exact solution uðtÞ ¼ Cð1 þ aÞ1 ðt ta Þ which exists for t b 1 and possesses the property (L). Theorem 1 demonstrates that the linear-like asymptotic behavior of solutions to Eq. (1) does not inevitably require strong integral restrictions on the functions hi ðtÞ. In fact, assumptions on coe‰cients imposed by Cohen [1], Constantin [2], Meng [9], Mustafa and Rogovchenko [10], S. Rogovchenko and Yu. Rogovchenko [12], Rogovchenko [13], Tong [15], Trench [16] and Waltman [18] are merely due to demonstration technique. 3.

Unbounded domains of initial data

In this section we are concerned with Eq. (1) under the assumption that the real-valued continuous function f ðt; u; u 0 Þ satisfies u ð27Þ j f ðt; u; u 0 Þj a aðtÞg u 0 : t

176


In what follows, we suppose that t b t0 b 1, the function aðtÞ is continuous, nonnegative and satisfies the growth condition ðy aðsÞ ds < þy for some a A ð0; 1Þ: a t0 s Furthermore, let gðsÞ be continuous, nondecreasing, positive for s > 0, and such that for every x b t0 xgðsÞ a gðx 1a sÞ;

ð28Þ

s b 0:

The class of functions satisfying these assumptions is non-empty. For instance, one may think of gðsÞ ¼ s 1=ð1aÞ , s b 0. Fix a c > 0. For u b c, we introduce the function GðuÞ by ðu ds def GðuÞ ¼ c gðsÞ and require that ðy t0

aðsÞ ds < GðþyÞ: sa

Note that by (28) the value GðþyÞ is finite. Indeed, let x ¼ x 1=ð1aÞ and s ¼ 1 in (28). Then ðy ðy dx dx 1a a=ð1aÞ 1 a ½gð1Þt0 ¼ < þy: 1=ð1aÞ a t0 gðxÞ t0 gð1Þx In order to establish the main result in this section, we need two auxiliary lemmas. Lemma 4. Let a; b > 0 be real constants, u 0 A R n be a given vector, def

U ¼ fu A R n : ku u 0 k a bg; a def V ¼ v A R n : kvk a ½b þ ku 0 k ; t0 and let the vector function g : D ¼ ½t0 ; t0 þ a U V ! R n be continuous. Then there exists a solution xðtÞ of the Cauchy problem ðt uðsÞ 0 u ðtÞ ¼ g t; uðtÞ; ds ; uðt0 Þ ¼ u 0 ð29Þ t0 s defined on ½t0 ; t0 þ a, where a ¼ minfa; bM 1 g, and M¼

sup ðt; u; vÞ A D

kgðt; u; vÞk:


177

The proof of this result is very similar to that of Peano’s theorem (see Kartsatos [5, Theorem 3.1, pp. 51–52]). Lemma 5. Assume that the vector function g : ½t0 ; t0 þ a R n R n ! R n is continuous and let u 0 A R n be a given vector. Let xðtÞ be a solution of the Cauchy problem (29) satisfying kxðtÞk < l for as long as it exists to the right of t0 , where l > 0 is a real constant. Then the solution xðtÞ is continuable up to the point t ¼ t0 þ a. The proof follows the same lines as in Kartsatos [5, Theorem 3.8, p. 57], and we use the definition of continuation of solutions to (29) given by Kartsatos in [5, Definition 3.5, p. 55]. Theorem 6. ð30Þ

Assume that ðy ðy aðsÞ du : ds < a 1a s gðuÞ t0 cþjz0 jt0

Then every solution uðtÞ of Eq. (1) satisfying initial conditions (7), where z0 is defined by (19), exists in the future. Furthermore, uðtÞ ¼ a u t þ oðtÞ and u 0 ðtÞ ¼ a u þ oð1Þ as t ! þy. Proof. We use the relationship between uðtÞ and zðtÞ established in Theorem 1. Then Eq. (1) can be written as z 0 ¼ t1 z f ðt; u; u 0 Þ;

t A ½t0 ; TÞ;

where uðtÞ is a solution of the Cauchy problem (1), (7) with the initial data satisfying (30) and zðtÞ ¼ u 0 ðtÞ t1 uðtÞ, t A ½t0 ; TÞ. Our aim is to prove the following two assertions. (i) Every solution of the Cauchy problem ð31Þ

z 0 ¼ t1 z þ aðtÞgðjzjÞ;

zðt0 Þ ¼ z0 ;

where z0 satisfies (30), exists in the future. (ii) For every solution zðtÞ in (i), there exists a real number l A ð0; 1Þ such that zðtÞ ¼ Oðtl Þ as t ! þy. To prove claims (i) and (ii), consider solution z0 ðtÞ of the Cauchy problem (31) defined for t A ½t0 ; TÞ. Clearly, z0 ðtÞ is the unique solution of z 0 ¼ t1 z þ aðtÞgðjz0 ðtÞjÞ;

zðt0 Þ ¼ z0 ;

on ½t0 ; TÞ. Using variation of constants method, we establish that z0 ðtÞ satisfies the integral equation ð z0 t 0 1 t z0 ðtÞ ¼ þ ð32Þ saðsÞgðjz0 ðsÞjÞds; t A ½t0 ; TÞ: t t0 t

178


Making use of (32) and assumption (28), we obtain for all t A ½t0 ; TÞ ð jz0 jt0 1 t ð33Þ t 1a jz0 ðtÞj a a þ a aðsÞ½sgðjz0 ðsÞjÞds t t0 t ð 1 t a jz0 jt01a þ a aðsÞgðs 1a jz0 ðsÞjÞds t t0 ðt 1 1a aðsÞgðs 1a jz0 ðsÞjÞds: a jz0 jt0 þ a t0 s Introducing the notation yðtÞ ¼ c þ t 1a jz0 ðtÞj; we deduce from (33) that yðtÞ a c þ jz0 jt01a þ

ðt

1 aðsÞgðyðsÞÞds; a s t0

t A ½t0 ; TÞ:

Thus, for all t A ½t0 ; TÞ Gð yðtÞÞ a Gðc þ

jz0 jt01a Þ

þ

< Gðc þ jz0 jt01a Þ þ

ðy t0

aðsÞ ds sa

ðy

du ¼ GðþyÞ: cþjz0 jt01a gðuÞ

Since the function G is invertible on ½0; GðþyÞÞ, we have ðy aðsÞ yðtÞ a G1 Gðc þ jz0 jt01a Þ þ ð34Þ ds ¼ K < þy: a t0 s It follows from (34) that jz0 ðtÞj a

K a K; t 1a

t A ½t0 ; TÞ;

which ensures that T ¼ þy, and thus z0 ðtÞ exists in the future. inequality (34) implies that for any l A ð0; 1 a z0 ðtÞ ¼ Oðtl Þ

Furthermore,

as t ! þy:

In a similar manner, one can establish the following two assertions. (iii) Every solution of the Cauchy problem ð35Þ

z 0 ¼ t1 z aðtÞgðjzjÞ;

where z0 satisfies (30), exists in the future.

zðt0 Þ ¼ z0 ;


179

(iv) For every solution zðtÞ in (iii), there exists a l A ð0; 1Þ such that zðtÞ ¼ Oðtl Þ as t ! þy. Coming back to Eq. (1), we conclude that for t A ½t0 ; TÞ 1 z 0 ¼ z f ðt; u; u 0 Þ t ðt ðt 1 u0 zðsÞ u0 zðsÞ ds ; þ ds þ zðtÞ : þ ¼ z f t; t t t0 t0 t0 s t0 s Thus, for every solution uðtÞ of the Cauchy problem (1), (7) defined on ½t0 ; TÞ, there exists a unique solution zðtÞ, also defined on ½t0 ; TÞ, of the Cauchy problem ðt zðsÞ u0 0 ds ; zðt0 Þ ¼ z0 ¼ u1 ; z ¼ V t; zðtÞ; ð36Þ t0 t0 s where 1 u0 u0 def V ½t; z; w ¼ z f t; t þw ; þwþz ; t t0 t0 and zðtÞ can be viewed as a nonhomogeneous term in Eq. (22). It follows from (27) that solution zðtÞ of (36) satisfies for t A ½t0 ; TÞ differential inequalities z 0 a t1 z þ aðtÞgðjzjÞ;

zðt0 Þ ¼ z0 ;

z 0 b t1 z aðtÞgðjzjÞ;

zðt0 Þ ¼ z0 :

and

If zmin ðtÞ and zmax ðtÞ are the minimal and maximal solutions of the Cauchy problems (35) and (31) on ½t0 ; þyÞ, one has ð37Þ

zmin ðtÞ a zðtÞ a zmax ðtÞ;

t A ½t0 ; TÞ:

Furthermore, there exists a l A ð0; 1Þ such that ð38Þ

zmin ðtÞ ¼ Oðtl Þ

and

zmax ðtÞ ¼ Oðtl Þ

as t ! þy:

Using (37), (38), and Lemma 5, we conclude the following. (v) Every solution zðtÞ of the Cauchy problem (36) exists in the future and is bounded. (vi) For every solution zðtÞ in (v), there exists a l A ð0; 1Þ such that zðtÞ ¼ Oðtl Þ as t ! þy. To complete the proof, one follows the same steps as in Theorem 1. 9

180


Remark 7. As it has been mentioned in the Introduction, nonlinear differential equation (10) has a non-extendable solution (11) defined on ½1; 2Þ. Simple computation of z0 for this solution gives def

z0 ¼ u 0 ð1Þ uð1Þ=1 ¼ 8=3: With the choice a ¼ 1=2 and aðtÞ ¼ t2 , t b t0 ¼ 1, condition (30) reads as 2=3 < ðc þ jz0 jÞ1 : Since c > 0 is arbitrary real number, the best possible estimate is jz0 j < 3=2. Initial data of the blowing up solution (11) do not belong to the region of linear-like behavior obtained in Theorem 6. This region is neither R 2 , nor the empty set since there are solutions satisfying jz0 j < 3=2 with linear-like behavior at infinity as, for instance, uðtÞ ¼ t=8. Example 8. ð39Þ

Consider the nonlinear di¤erential equation 1 u 2 3tu 2t 2 u 0 00 0 u u 1þ ¼ 0; t b 1: t 8ðt 2 þ u 2 Þ 15t 3=2

A straightforward computation yields ! 2 2 1 1 u 0 ð3uÞ=ð2tÞ 1 3=2 0 u 3=2 0 u t u 1 a t u t 15 t 4 1 þ ðjuj=tÞ 2 15 2 0 u 3 1 3=2 1 u þ 1 t3=2 ðjuj=tÞ u 0 u t 2 2 60 t 120 t 1 þ ðjuj=tÞ 1 þ ðjuj=tÞ 2 ! 0 u 3 7 3=2 0 u t a u t þ u t : 80 þ

Therefore, aðtÞ ¼ ð7=80Þt3=2 , gðsÞ ¼ s 2 þ s 3 , t0 ¼ 1, and a ¼ 1=2. ð40Þ ð41Þ

ðy

7s3=2 7 ds ¼ ; 1=2 80 1 80s ðy ðy ds ds 1 b ¼ : 3 2 3 2ðc þ 31 þ jz0 jÞ 2 cþjz0 j s þ s cþjz0 j ðs þ 31 Þ

Solving the inequality 7 1 1 < ; 80 2 ðc þ 31 þ jz0 jÞ 2

Then


181

we conclude that the region of linear-like behavior contains the set pffiffiffiffiffiffiffiffiffiffi jz0 j < 40=7 31 A 2:05723: ð42Þ Thus, all solutions of Eq. (39) with initial data satisfying (42) exhibit desired behavior at infinity. Since we approximate the integral (41) instead of computing it exactly, the estimate (42) is not sharp and can be improved. It follows from the inequality ðy cþjz0 j

s2

ds 7 a ðc þ jz0 jÞ1 a 3 80 þs

that one can find solutions of Eq. (39) with di¤erent from linear asymptotic behavior as t ! þy, if any, in the set containing initial data satisfying jz0 j > 80=7, although the description of this set is not precise since we only estimated (41). In fact, Eq. (39) has the exact solution uðtÞ ¼ 45t 3=2 whose derivative grows without limit at infinity, and a simple computation shows that for this solution z0 ¼ 45=2. This example confirms coexistence of positive half-trajectories with qualitatively di¤erent asymptotic behavior discussed by the present authors in [10] for the general equation (1). Even for particular classes of equations studied in this paper, one can have, for instance, both linear-like solutions with bounded derivatives and solutions having unbounded derivatives, which reflects complicated nature of the problem. 4.

Discussion

We would like to conclude the paper pointing out open problems and indicating possible applications and extensions of the results. Remark 9. For Eq. (12), the coe‰cient hðtÞ in the inequality (4) is aðtÞ. The only conditions imposed on it are continuity and nonnegativity, there are no integrability assumptions or restrictions on its asymptotic behavior. We believe that these hypotheses are minimal for this class of equations and cannot be relaxed. It is not possible to achieve a similar goal for the general nonlinear di¤erential equation (1). Our e¤orts aimed on relaxing integrability conditions on the coe‰cients imposed by most authors, see Cohen [1], Constantin [2], Kusano and Trench [7, 8], Meng [9], Mustafa and Rogovchenko [10], S. Rogovchenko and Yu. Rogovchenko [12], Rogovchenko [13], Tong [15] and Trench [16], proved to be successful for Eq. (12). However, the problem remains open for other classes of nonlinear di¤erential equations. Remark 10. As opposed to related results reported in the literature, the region of linear-like behavior in the ðu; u 0 Þ-plane defined by the condition (30)

182


contains unbounded strip. Study of the relationship between unbounded connected domains of R 2 having smooth boundary and linear-like solutions of nonlinear di¤erential equations, if any, is another interesting open problem. Remark 11. Following Kusano and Trench [8, Example 2] and Zhang [19, Corollary 3], let us focus attention on the role played by the terms u=t and u 0 u=t in Eqs. (1) and (12) in connection with the problem of existence of radial solutions to nonlinear elliptic equations. Consider equation ð43Þ

su þ f ðjxj; u; j‘ujÞ ¼ 0;

x A Wt0 ;

where Wt0 ¼ fx A R 3 : jxj > t0 g. If u ¼ uðjxjÞ is a radially symmetric solution of Eq. (43), the function yðtÞ ¼ tuðtÞ satisfies the ordinary di¤erential equation y 1 0 y þ tf t; ; y t t 00

y ¼ 0; t

t > t0 :

In this case, we have su ¼ t1 y 00 ðtÞ;

u 0 ¼ t1 zðtÞ

and

j‘uj ¼ t1 jzðtÞj;

where zðtÞ ¼ y 0 ðtÞ t1 yðtÞ. It is straightforward now to restate Theorem 6 for Eq. (43). Finally, we observe that property (L) can be generalized as follows: for a continuable solution uðtÞ of Eq. (1), there exists a real constant a such that uðtÞ lim u 0 ðtÞ ð44Þ ¼ a: t!y t It is easy to see that (44) implies lim

t!y

uðtÞ ¼ a: t ln t

We believe that the general condition (44) is of interest for the study of the problems where property (L) has been considered (for instance, nonoscillation of solutions of ordinary di¤erential equations or asymptotic behavior of solutions of elliptic equations in exterior domains). To the best of our knowledge, no attempts in this direction have been made yet. To show the potential of this generalization, consider a perturbation of Eq. (1), namely, nonlinear di¤erential equation ð45Þ

u 00 þ f ðt; u; u 0 Þ ¼ bðtÞ;

t b t0 ;


183

where bðtÞ is a continuous function satisfying lim

t!y

1 t

ðt

sbðsÞds ¼ a A R:

t0

Elementary examples of such functions are, for instance, bðtÞ ¼ t1 and bðtÞ ¼ t1 ð21 cos tÞ, where t b t0 b 1. Assume, in addition, that u j f ðt; u; u 0 Þj a hðtÞu 0 ; t where hðtÞ is a continuous, nonnegative function such that ð46Þ

ðy

shðsÞds < þy:

t0

Theorem 12. Let u 0 A R. Then there exists a solution uðtÞ of the problem (45), (23), defined on ½t0 ; þyÞ, which satisfies uðtÞ 0 lim u ðtÞ ¼ a: t!y t Proof. Let uðtÞ be a solution of (45), (23). Using the transformation v ¼ tu 0 u, we obtain ðt v 0 þ g t; vðtÞ; s2 vðsÞds ¼ tbðtÞ; t b t0 ; t0

where u0 u0 v gðt; v; wÞ ¼ tf t; t þw ; þwþ : t t0 t0 We have ðt g t; vðtÞ; s2 vðsÞds a hðtÞjvðtÞj: t0

Introduce the space Lðt0 Þ of all continuous real-valued functions vðtÞ defined on ½t0 ; þyÞ with the property lim

t!y

vðtÞ A R: t

Being endowed with the Chebyshev norm

184


kvk ¼ sup tbt0

jvðtÞj ; t

Lðt0 Þ becomes a Banach space. Define the integral operator T : Lðt0 Þ ! Lðt0 Þ by the formula ðt ðy ðs ðTvÞðtÞ ¼ sbðsÞds þ g s; vðsÞ; t2 vðtÞdt ds; t b t0 : t0

t

t0

Then, using the technique developed by the authors in [10], we conclude that T is completely continuous and has a fixed point v0 ðtÞ in Lðt0 Þ. Let ðt u0 2 uðtÞ ¼ t þ s v0 ðsÞds : t0 t0 Since v0 ðtÞ is sublinear in the sense that jv0 ðtÞj a kv0 kt for all t b t0 , ðt ðt 0 u ðtÞ uðtÞ 1 sbðsÞds ¼ ðTv0 ÞðtÞ 1 sbðsÞds t t t t t0

t0

a kv0 k

1 t

ðy shðsÞds;

t b t0 :

t

Passing to the limit as t ! y, we complete the proof of the theorem.

9

Remark 13. Clearly, a ¼ 0 whenever the function tjbðtÞj is integrable, and one can use the technique developed by the authors in [10, Lemma 7] to prove that assumptions (4) and (6) hold for Eq. (45) with the choice pi ðwÞ ¼ w þ 1=2, i ¼ 1; 2. According to [10, Theorem 6], for any pair of real numbers A; B there exists a solution uðtÞ of Eq. (45) defined on ½t0 ; þyÞ with the asymptotic representation uðtÞ ¼ At þ B þ oð1Þ as t ! þy. This result and the fact that for a ¼ 0 condition (44) does not provide enough information on the behavior of uðtÞ for large t stimulate particular interest to the case a 0 0. In fact, assume that a > 0 and bðtÞ b 0 for all t b t0 . Then ðy ð47Þ bðsÞds ¼ þy: t0

To prove (47), note that 1 0 < a ¼ lim t!þy t

ðt

1 sbðsÞds ¼ lim t!þy t t0

ðt

ðy sbðsÞds a

x

bðsÞds x

for all x b t0 , which yields the conclusion. Condition (47) does not allow application of the results reported by the authors in [10], emphasizing thus importance of Theorem 12.


185

Acknowledgements. The authors are deeply indebted to the Referee for careful and thoughtful reading of the manuscript and many useful suggestions which helped to improve the presentation of results. O.M. thanks Professor Valerian Astefanei of the University of Craiova for valuable advices. Yu.R. gratefully acknowledges financial support and warm hospitality of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy which facilitated the work on this paper. References [ 1 ] Cohen, D. S., The asymptotic behavior of a class of nonlinear di¤erential equations, Proc. Amer. Math. Soc., 18 (1967), 607–609. [ 2 ] Constantin, A., On the asymptotic behavior of second order nonlinear di¤erential equations, Rend. Mat. Appl., 13 (1993), 627–634. [ 3 ] Dannan, F. M., Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behavior of certain second order nonlinear di¤erential equations, J. Math. Anal. Appl., 108 (1985), 151–164. [ 4 ] Fukagai, N., Existence and uniqueness of entire solutions of second order sublinear elliptic equations, Funkc. Ekvac., 29 (1986), 151–165. [ 5 ] Kartsatos, A. G., Advanced Ordinary Di¤erential Equations, 2nd ed., Mancorp Publishing, Inc., Tampa, Florida, 1993. [ 6 ] Kusano, T., Naito, M., and Usami, H., Asymptotic behavior of solutions of a class of second order nonlinear di¤erential equations, Hiroshima Math. J., 16 (1986), 149–159. [ 7 ] Kusano, T., and Trench, W. F., Global existence theorems for solutions of nonlinear differential equations with prescribed asymptotic behavior, J. Lond. Math. Soc., 31 (1985), 478– 486. [ 8 ] Kusano, T., and Trench, W. F., Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary di¤erential equations, Ann. Mat. Pura Appl., 142 (1985), 381–392. [ 9 ] Meng, F. W., A note on Tong paper: The asymptotic behavior of a class of nonlinear di¤erential equations of second order, Proc. Amer. Math. Soc., 108 (1990), 383–386. [10] Mustafa, O. G., and Rogovchenko, Yu. V., Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear di¤erential equations, Nonlinear Anal., 51 (2002), 339–368. [11] Nehari, Z., On a class of nonlinear second-order di¤erential equations, Trans. Amer. Math. Soc., 95 (1960), 101–123. [12] Rogovchenko, S. P., and Rogovchenko, Yu. V., Asymptotic behavior of certain second order nonlinear di¤erential equations, Dynam. Systems Appl., 10 (2001), 185–200. [13] Rogovchenko, Yu. V., On the asymptotic behavior of solutions for a class of second order nonlinear di¤erential equations, Collect. Math., 49 (1998), 113–120. [14] Rogovchenko, Yu. V., and Villari, Gab., Asymptotic behavior of solutions for second order nonlinear autonomous di¤erential equations, Nonlinear Di¤erential Equations Appl., 4 (1997), 271–282. [15] Tong, J., The asymptotic behavior of a class of nonlinear di¤erential equations of second order, Proc. Amer. Math. Soc., 84 (1982), 235–236. [16] Trench, W. F., On the asymptotic behavior of solutions of second order linear di¤erential equations, Proc. Amer. Math. Soc., 14 (1963), 12–14.

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[17] Usami, H., Global existence and asymptotic behavior of solutions of second-order nonlinear di¤erential equations, J. Math. Anal. Appl., 122 (1987), 152–171. [18] Waltman, P., On the asymptotic behavior of solutions of a nonlinear equation, Proc. Amer. Math. Soc., 15 (1964), 918–923. [19] Zhang, Y., Positive solutions of singular sublinear Dirichlet boundary value problems, SIAM J. Math. Anal., 26 (1995), 329–339. nuna adreso: Octavian G. Mustafa Centre for Nonlinear Analysis University of Craiova Romania Yuri V. Rogovchenko Department of Mathematics Eastern Mediterranean University Famagusta, TRNC, Mersin 10 Turkey E-mail: [email protected] (Ricevita la 23-an de au˘gusto, 2002) (Reviziita la 23-an de oktobro, 2003)